Product integral

The expression "product integral" is used informally for referring to any product-based counterpart of the usual sum-based integral of classical calculus. The first product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations.[1][2] (Please see "Type II" below.) Other examples of product integrals are the geometric integral ("Type I" below), the bigeometric integral, and some other integrals of non-Newtonian calculus.[3]

This article adopts the "product" ∏{\displaystyle \prod } notation for product integration instead of the "integral" ∫{\displaystyle \int } (usually modified by a superimposed "times" symbol or letter P) favoured by Volterra and others. An arbitrary classification of types is also adopted to impose some order in the field.

It is very useful in stochastics where the log-likelihood (i.e. the logarithm of a product integral of independent random variables) equals the integral of the log of these (infinitesimally many) random variables:

Under these definitions, a real function is product integrable if and only if it is Riemann integrable. There are other more general definitions such as the Lebesgue product integral, Riemann–Stieltjes product integral, or Henstock–Kurzweil product integral.

which is not a multiplicative operator. (So the concepts of product integral and multiplicative integral are not the same). The Volterra product integral is most useful when applied to matrix-valued functions or functions with values in a Banach algebra, where the last equality is no longer true (see the references below).